Series of Functions and Power Series

Uniform Convergence and Properties of Limit Functions

TipUniform and Pointwise Convergence

Let \(I \subseteq \mathbb{R}^m\), for a function \(f : I \to \mathbb{R}\), we denote:

\[ \|f\|_\infty = \sup_{\mathbf{x} \in I} |f(\mathbf{x})|. \]

For a sequence of functions \(f_n : I \to \mathbb{R}\), we say that \(f_n\) uniformly converges to \(f\) on \(I\) (denoted as \(f_n \overset{I}{\rightrightarrows} f\)), if:

\[ \lim_{n \to +\infty} \|f_n - f\|_\infty = 0, \]

That is: for any \(\varepsilon > 0\), there exists an \(N(\varepsilon)\) such that for any positive integer \(n > N(\varepsilon)\) and for any \(\mathbf{x} \in I\), \(|f_n(\mathbf{x}) - f(\mathbf{x})| < \varepsilon\). Here, “uniform” means that \(N(\varepsilon)\) is independent of \(\mathbf{x} \in I\).

If for any \(\mathbf{x} \in I\), we have \(\lim_{n \to +\infty} f_n(\mathbf{x}) = f(\mathbf{x})\), then \(f_n\) is said to pointwise converge to \(f\) on \(I\), which means for any \(\mathbf{x} \in I\) and any \(\varepsilon > 0\), there exists an \(N(\mathbf{x}, \varepsilon)\) such that for any positive integer \(n > N(\mathbf{x}, \varepsilon)\), \(|f_n(\mathbf{x}) - f(\mathbf{x})| < \varepsilon\).

A sequence of functions \(f_n\) is said to converge non-uniformly to \(f\) on \(I\) if \(f_n\) converges pointwise to \(f\) on \(I\), but does not converge uniformly to \(f\).